Where Have All the Theorems Gone?

I enjoyed reading Euclid last month. In my eighth annual meeting with the Alexandrian mathematician, I reached the section on solid geometry and learned the word parallelepiped. But my post today has much less to do with book XI of the Elements than it does with stories of a handful of geometrical encounters from my life.

A few years ago, a friend asked if I could tutor his son in geometry. I’m your guy, I told him. I loved geometry in high school. I loved the beauty of the logic, the practicality of the conclusions. I loved the way it felt to comprehend the proof of a proposition that did not at all seem clear on first acquaintance. I loved the grand view of an edifice of knowledge built by meticulous method upon the foundation of just a few unprovable axioms. My high-school trigonometry teacher mistakenly handed me a geometry test one day. When I pointed his mistake out to him, he told me to go ahead and complete the exam, offering to give me the grade if I made a 100% on it. And so I did. Yeah. I would be happy to tutor my friend’s son.

So he showed up with his large textbook, the word GEOMETRY standing monumentally atop the front cover in large, white, bold, sans-serif characters. I smiled with anticipation, opening the book and flipping through its pages, ready to breathe in the vibrant airs arising from the comforting marked diagrams of arcs and line segments and pacifically indented tables of premises and justifications. Instead, I met a vacuum and gasped. I saw a picture of a smiling student with a calculator, but no triangles with alphabetically indexed points. I found plenty of subheadings and colored sidebars, but no proofs, no lists of theorems. The book taught (I use that word loosely) some algebra and some number theory, but no geometry. Admittedly, one section of the text offered formulae for the areas of various plane figures, but no logical demonstration why a triangle’s area should depend on base and height regardless of the angles involved. So the school offered a class called “Geometry,” and the school used a book called “Geometry,” and the school wrote the word “Geometry” on transcripts, but the school didn’t actually teach geometry.

We’ll jump fifteen years or so, all the way up to this just-ended spring semester. In a graduate course on the history of music theory, my students discovered that music scholars in the Middle Ages defined pitch by dividing the length of a string into various numbers of equal parts. We had earlier covered the medieval curriculum of Trivium – grammar, logic, and rhetoric – and Quadrivium – arithmetic, geometry, music, and astronomy. I asked my students which of these seven liberal arts the boys in the monastery schools used to divide the string evenly, thinking of it as an easy question with an obvious answer, designed not to quiz them but merely to provide a moment of interaction and to recapture their attention. But the room was silent. “Well, OK,” I improvised. “You remember doing this in geometry class, right?” They answered my second question with more silence, a couple of nods, and a lot of confused faces. After I told this story to some friends that evening, they reminded me of that high-school textbook, and I realized that many of my graduate students probably took a class called Geometry without ever learning how to divide a line into a number of even parts. Telling them that medieval boys learned “geometry” had conveyed no information. I could have substituted a random Sanskrit word and had the same effect.

Faced with my failure to communicate, I had two choices. I could rail against the times like Canute commanding the waves. Or I could teach some geometry. So for the last two days of class, I brought in a compass, a straight edge, and my copy of Euclid, and led them along an abbreviated path from book I, proposition 1 to book VI, proposition 9: To cut off a prescribed part from a given straight line. Did they learn any geometry? Did they learn about proof and self-evident axioms? Did they learn why music theory was considered a mathematical art in the eleventh century? I don’t know. I think they learned something about me.